Introduction
Value at Risk (VaR) is a key indicator in financial risk management, quantifying the maximum potential loss on a portfolio over a given period at a specified confidence level. For example, a 95% 1-day VaR of €1 million means there's a 5% chance of losing more than that in one day.
Why is it crucial in 2026? With increased market volatility from geopolitics, crypto, and AI trading, regulators like Basel III require precise VaR for banks and funds. This intermediate tutorial, without code, focuses on solid theory: from foundations to advanced methods. You'll learn to interpret VaR like a pilot reads an altimeter—not a absolute guarantee, but a vital warning signal. Through concrete examples (equity-bond portfolios), we progress to pro best practices, avoiding pitfalls that sank institutions like LTCM in 1998. Result: actionable mastery for your risk reports or audits (128 words).
Prerequisites
- Statistics basics: mean, standard deviation, normal distribution.
- Finance knowledge: returns, volatility, correlations.
- Familiarity with Excel or analytics tools for manual simulations.
- Intermediate level: conditional probabilities and normal distribution.
- Estimated time: 2 hours of active reading.
Step 1: Understand VaR Foundations
Precise definition: VaR measures the quantile loss of a portfolio. General formula: VaR_α = -inf {x | P(L > x) ≤ 1-α}, where L is the loss, α the confidence level (e.g., 95% or 99%).
Horizon and confidence: Typically 1 day/10 days, 95%/99%. Analogy: like a safety net at 95%—it catches 95% of falls, but not the extreme 5%.
Concrete example: €10M portfolio (60% CAC 40 stocks, 40% government bonds). 2023-2025 history: stock volatility 18%, bonds 4%, correlation -0.2. At 95%/1 day, VaR ≈ €250k (expected max loss).
Interpretation: Not an expected loss (that's ES), but a probabilistic threshold. Use it in backtesting: compare realized vs. predicted VaR over 250 days to validate (failure rate <5% at 95%).
Step 2: Parametric Method (Variance-Covariance)
Principle: Assumes normal returns. VaR = Z_α × σ_p × V_p, where Z_α=1.65 (95%), σ_p=portfolio standard deviation, V_p=value.
Matrix calculation: For n assets, σ_p = √(w' Σ w), Σ=covariance matrix, w=weights.
Detailed example: Above portfolio. σ_stocks=0.18/√252≈1.13%/day, σ_bonds=0.04/√252≈0.25%/day. Cov=ρσ1σ2=-0.2×1.13%×0.25%≈-0.06%. σ_p=√(0.6²×1.13² + 0.4²×0.25² + 2×0.6×0.4×(-0.06%))≈0.75%/day. VaR95=1.65×0.75%×10M€=€123k.
Advantages: Fast, analytical. Limitations: Ignores fat tails (kurtosis>3). Test normality with Jarque-Bera (stat>5.99 rejects H0).
Validation checklist:
- Check returns stationarity (ADF test p<0.05).
- Recalculate rolling covariances (60-day window).
Step 3: Historical Method
Principle: Non-parametric. Sort historical losses (e.g., 1000 days), take α quantile.
Precise steps:
- Collect daily P&L returns over T=500-1000 days.
- Calculate P&L = -ΔV portfolio.
- Sort ascending, VaR_95% = 5th worst loss.
Example: Over 756 days (3 years), sorted losses: -500k, -400k, ..., +2M. 5% quantile = -€280k (38th worst).
Advantages: Captures non-normality, fat tails. Drawbacks: Sensitive to periods (2008 crisis inflates VaR).
Improvements:
- EWMA weighting (recent weights ×λ=0.94).
- Bootstrap: resample 1000x historicals for VaR CI [±20% typical].
Case study: In 2020 COVID, historical VaR (2017-2019 window) underestimates crash (-40% vs. actual -25%), hence need stress tests.
Step 4: Monte Carlo Method
Principle: Stochastic simulations. Generate 10k-100k future scenarios via models (GBM, GARCH).
Detailed workflow:
- Model factors (SDE dS=μdt+σdW).
- Simulate paths (antithetic for 50% variance reduction).
- Calculate P&L per scenario, take α quantile.
Advanced example: Portfolio with options. Heston model (stochastic vol). 50k runs: VaR99%/10 days=€1.2M (uncertainty ±15%).
Advantages: Flexible (non-linearities, jumps via Merton). Cost: Compute-intensive (but cloud-friendly in 2026).
Validation: KS-test simulated vs. historical distribution (D<0.05). Calibrate params via MLE on recent data.
Analogy: Like a virtual crash-test of 10k cars vs. one real historical accident.
Step 5: Comparison and Backtesting
Comparison table:
| Method | Speed | Tail Accuracy | Example VaR95%/1 day |
|---|---|---|---|
| -------- | ------- | --------------- | ----------------------- |
| Parametric | +++ | -- (normal) | €123k |
| Historical | ++ | ++ (data-driven) | €280k |
| MC | - | +++ (modeled) | €180k |
Method choice: Parametric for liquid assets, historical for illiquid, MC for derivatives. Hybrid historical+MC standard in 2026.
Best Practices
- Daily recalculation with tick-level data for intraday VaR.
- Incorporate dynamic correlations (DCC-GARCH): CAC 40/Bund correlation shifts from -0.3 to 0.8 in stress.
- Combine with ES/CVaR: ES= E[L| L>VaR], +20-50% over simple VaR.
- Stress and scenarios: VaR + 5 scenarios (recession, geo) for FRTB.
- Audit documentation: Log params, data sources, stat tests (p-values).
Common Mistakes to Avoid
- Blind normality assumption: Markets have kurtosis 10+ (vs. 3 normal) → underestimates VaR x2 (LTCM 1998).
- Fixed historical window: Ignores regimes (2022 vol spike) → use rolling/adaptive.
- Ignore liquidity: Mark-to-market VaR misses bid-ask → +30% in crisis.
- Isolated confidence levels: Always pair 95%/99%, separate backtests.
Next Steps
- Books: Risk Management by McNeil (advanced models).
- Tools: RiskMetrics (JPM), @Risk Excel.
- Regulations: Basel III/FRTB (VaR vs. ES).
- Check out our Learni courses: Quantitative Risk Management – hands-on VaR/ES workshops.
- Articles: Jorion Value at Risk (3rd ed.), SSRN 2025 papers on ML-VaR.